Displacement–length scaling in three dimensions: the importance of aspect ratio and application to deformation bands

Deformation bands confined to a 9-m-thick layer of the Entrada Sandstone in Utah accumulate less displacement per unit length than fractures that are not stratigraphically confined. This difference in displacement–length (D–L) scaling is related to the increasing length-to-height (aspect) ratio of the bands. Here we derive new expressions for displacement–length scaling and fracture strain for three-dimensional (3-D) elliptical fractures. The maximum (relative) displacement Dmax on an elliptical fracture surface depends on the fracture geometry (both length L and height H), the end-zone length (through driving stress and the rockʹs yield strength), and the properties of the surrounding rock (modulus, Poissonʹs ratio). A given elliptical fracture will show different values of Dmax/L in horizontal and vertical sections due to differences in fracture dimension (length vs. height) and end-zone length. A population of elliptical fractures can accommodate less displacement or strain if fracture aspect ratios increase with L than a population of fractures having constant aspect ratios. These relationships reveal how 3-D fracture geometries systematically influence the population statistics. The magnitude of horizontal (extensional) fracture strain accommodated by the population of deformation bands predicted by the analysis is consistent with that obtained independently from traverse measurements on the outcrop (0.12%). The 3-D fracture geometry can contribute at least an order of magnitude in displacement deficit (or excess) relative to tall 2-D fractures and comparable scatter on maximum displacement vs. length (D–L) diagrams. In general, fractures confined to stratigraphic layers grow nonproportionally (L/H≠constant for L>the layer thickness), leading to reduced capacity to accommodate displacement and a shallower slope on the D–L diagram. Similarly, fractures that grow by segment linkage (preferentially down-dip or along-strike) scale as nonproportional 3-D fractures. A unit slope on D–L diagrams implies proportional growth (L/H=constant). Faults with slip surfaces and other fractures with nonpreferred growth directions can produce unit slopes, so that particular trajectories on D–L diagrams can reveal physical controls on fracture growth, such as stratigraphic confinement or segment interaction and linkage.